Gregory G. Smith

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

Multigraded Castelnuovo-Mumford regularity

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.

Uniform bounds on multigraded regularity

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann’s regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert

SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m 1 1 ⊗ � � � ⊗ B ml l . We give conditions on the mi which guarantee that the ideal of X in P(H 0 (X, L) � ) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

Sums of squares and varieties of minimal degree

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree. This substantially extends Hilberts celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes Q for which every nonnegative Laurent polynomial with support contained in 2Q is a sum of squares.

Computing global extension modules

Abstract Let X be a projective scheme and let M and N be two coherent O X -modules. Given an integer m , we present an algorithm for computing the global extension module Ext X m ( M , N ). As a consequence, one may calculate the sheaf cohomology H m ( X , N ) and construct the sheaf corresponding to an element of the module Ext X 1 ( M , N ). This algorithm depends only on the computation of Grobner bases and syzygies and has been implemented in the computer algebra system Macaulay2.

Smooth and irreducible multigraded Hilbert schemes

Abstract The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z [ x , y ] , which establishes a conjecture of Haiman and Sturmfels.

Irreducible components of characteristic varieties

Abstract We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of the Weyl algebra A n . This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive. More explicitly, we consider a filtration of A n induced by any vector ( u , v )∈ Z n × Z n such that the associated graded algebra is a commutative polynomial ring. Any finitely generated left A n -module M has a good filtration with respect to ( u , v ) and this gives rise to a characteristic variety Ch ( u , v ) (M) which depends only on ( u , v ) and M . When ( u , v )=( 0 , 1 ) , the characteristic variety is involutive and this implies that its irreducible components have dimension at least n . In general, the characteristic variety may fail to be involutive, but we are still able to prove that each irreducible component of Ch ( u , v ) (M) has dimension at least n .

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bun dle is defined by the 2 � 2 minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions fo r such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.

D-Modules on Smooth Toric Varieties

Abstract Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal b . Let A denote the ring of differential operators on Spec(S). We show that the category of D -modules on X is equivalent to a subcategory of graded A-modules modulo b -torsion. Additionally, we prove that the characteristic variety of a D -module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic D -modules correspond to holonomic A-modules.